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Full text of "The Calculation of Compressible Flows with Local Regions of Supersonic Velocity"

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No. 1114 



By B. Gothert and K. H. Kawalki 


Berechnimg kompressibler Stromungen mit 
brtlichen tjberschallfeldern 

Forsch-ungsbericht Nr. 1794 


1 - Washington 


\ March 1947 

3 1176 014412259 

"SAnmasiJiL adtjecuri caisxsnuEE for jqsrgkautics 


By B. Gothart and. K, E. EamUd 

ABSTRACT: Xbe follovlng report lo ooxLcenxod vLth a metbod for 
the approxlmato oalculatlon of comproasl'ble flows 
abont pznflloB vlth locel regions of supersonic 
velocity. Tlw flow oi'ounfl. a slender profile Is 
treated as oil exampll.e. 

OIMJHE! I. Startenent of the PrpWLam. 

H. Stirvey of the I^i^thod used. 

III. Colculatloaa of the Bieurrple. 

17. Appro3dinate Treaiaiont of Local BpglonB of Supernonlo 

V, Synanoti'loal and Unarnsnetrlcal Koslone of Buper sonic 

VI. Summary. 


Sevei<al methods ore Tmovn tor the calculation of ccmpresslhle 
flovs at hl£^ subBonlc velocities. The resulting approximate 
BolutlonB are quite -osefol as long aa sound velocity is not 
exoeeded at any point of the flov field. Hovever, apparently all 
these approximate cELLoulations, t/lthout exception, cease to 
converge or to render useful flov pattozna If the condition of 
purely suhsonlc flov Is no longer satisfied. Moreover, numerous 
tests In vlnd tunnels conf limed the result that the reconversion 

*"Berechnung JcdnpresBlhler StrflBrunson mit b'rtllchen Uber- 
echallfeldem." Zentrale f(£r vlssenechaftllches Berlchtswesen 
der Luftfahrtforschung des Generalluftzeugmslsters {2XIB) Berlln- 
Adlerahof , Forechungaherlcht JTr. 179'^> Berlln-Adlershof, den 
7. August 19'+3. 

DACA IK So. llll^ 

of local supareoQlo flovB Into subsonic flovs la real flovs 
takSB place not steeuUlT- Imt generally "by means of a oocoipreSBlon 
slioclc which completely cbenf^s the vdiole flov pattezn^^ There- 
fore, the queetion eriees Aether a contlnuouB process from 
supersonic to subsoolo flov Is at all possllxLe for "bodies In 
parallel flow, even thoue^i the foxmatlon of a "bovasdory layer on 
the surface of the "body Is at first neQlected. 

An appyoxlBifltlon me-Qiod vas tested In the JfTL In order to 
olaarlfy these questions; this nusthod maJces possible the calculation 
of flows with local regions of supersonic velocity. 


The devolopnent of the method started frcm the fact' that the 
Vnovn approximation msthodo for ccmpreselhls flows, tdilch are 
without oiooptlon "based oa a step-by- step Improvecent of tlie 
Inccoipresslble flov, are quite appropriate for tho llTolted liomaln 
of the pure suboonlo flov eztendlnf; Srcsa. iJiflnlty to the sonic 
voloolty boundary In the flov field near tiie body. Eovever, 
for the region of the local supersonic ilov a method based on 
the pxt^pertles of s'lpersonlc flov will be subsequently used, as 
for Instonco the method of characteristics of Prandtl-Busemazin. 
Accordingly the partial areas of supersonic and subsonic flow, 
respectively, are calculr.ted SAparately by different methods 
\dilch In each case arc treated according to the peculiarities of 
the partial flow to be calculated. The partial flows that wore 
thus detezmlned must then 'bo Joined in such a manner that the 
flows agree on the surface of contact of the two regions, that is, 
on tho sonic volocitj" boundary, with respect to magnitude as well 
as to direction of the velocity. Figure 1 shove a scheoamtic 
representation of the boundary between the two flov areas. 

1. Subsonic Region with analT Perturbation 7eloolties 

Frandtl's rule will repi-esent a good approximation for a 
great part of the outer subsonic flov area, eijooe the perturbation 
velocities caused by the profile are sufficiently small up to 
Bcaae distance from the sonic boundary so that the assunrptiona of 

no account will be taken here of flows -idilch ali^tlj' exceed 
sound velocity because for them there is no certain way of 
distingui^lng between a steady trsnsJtlon and a cccrpresslon 
shock; neither by pressure -distribution measureoaents nor by optical 
observations, for Instance according to the schlleran method. . 

KAGA mso, mk 

FraDcLtl's approximation are rather veil aatlsfled. The Bubsoolc 
flov of this regloa-can, of cotErse, not "be o'btalned "by Blnsply 
dlstorttog the Inoampresslhle flov around the profile aocordlng 
to PrQiadtl*B rule. First, the local siiper sonic velocity field 
requires essentially more spaoe than the suhsonlc flov. The 
outer subsonic streamlines, therefore, ore vldened outvazrL not 
only "by the hody In the flov tut also hy the additional displace- 
ment due to the super sonla reloclty field. "Eoverer, this additional 
vldenlng of the streazallnoa ly the aupersonlo Telocity field can 
only affect the subsonic flov like a modification of the Isoundary 
cooodltlons at the sonic Telocity houndary; such a modification 
may he represented In a single way hy source end sink distrlhutlons, 
dlpoles, end so forth. 

2. Suhsonlo Eoglon vith El^ Porturhrtion Telocltles 

As mantloDod "before, the pMrt of the siibsonlc flow vfaich has 
t^ Tue calculated according to Pi'andtl does net extend as foi^ as 
the sonic houndary; It only reachos up to a hounlrtry lino near 
the sonic houndory determined "by an acreod siifflclontly small 
Talue of the perturhatlcn Tol^oltles. (Tlila houndorj'- line is 
represented In fig. 1 hy a datfied lino.) TIio calciilation for 
the rejjlon trcm this line to the sonic Telocity 'bouadary must 
generally "be csLLTlod out hy an ImproTed suhscnic method. A 
nunorlcal method of calculation seems to he particularly appro- 
prtate for this Inteimndiate region. The oiitor flow according 
to Prandtl, which io assumed as known, nujy "by tiio mothod hy 
continued a little further, namely up to the sonic Telocity 

One assumes, for Instance, that the flow field is coTered "by 
a rectangular grid of selected points. The Telocity cor^ononts 
T^: and Ty at these points in the outer field are knotm for 
each case according to. Prandtl* s approximation. Then the 
Telocltles at the inner grid points may he calculated fixim the 
Imown Taluss at the outer grid points if the differential 
quotients vlilch are declslTe for this continuation are appro- 
ximated "by the corresponding difference quotients.^ Figure 2 
shall "be considered as an ex&mple. Magnitude and direction of 
the Telocltles at all grid points outside of the "boundeury lines 
are assumed as known. The air density shell he known also. 
Then the exact equations of the continuity and the Irrotationality 
are, for the indicated grid point "2"! 

21 wish to express my thanks here to Dr. H. Schu"bert/D7L 
for his suggestion to treat ccmpresslhle flc/'s according to 
the difference method. 

HAGA m No. llll^ 


or, In the notation of difference giiotlentEi! 


^^3 ■ ^^1 Va ■ V ■ ' 



If one asBranes the grid to "be qijadratlc vlth neahes of equal vldth 
then ^ a ZSy and the equatlone gl7en aliove are eliapllf led as 

'X2 ^ ^A,k * P2 ^ ^^r.A,k ■ -V2 ^ ^^,1 - P2 ^ ^^73,1 (1*) 

"^^A,!. - ^^X3,i (It) 

Frcm the second eqiiatlon one Immediately obtained the Telocity 
component y-.. at the point A, -while the first equation gives 

a relation hetvean the velocity ccairponont v and the air 

density at the point A. Velocity and olr density are connected 
In a rather ccanplicated way hy the adlahatlc relation 



(p heire represents the eJLr density In the gas at rest a the 
critical Bo-imd velocity) . ITherefore, It will he praotloal to 
use the linearized relation Ap ■ j^ Av instead of the esaot 

ftmctlon, the error "beinc nogllgihle, heoanse of the snuQl width 
of the meshes: 

HACA TM So. mh 

l(p/Po) _-2,^. /pNi 

— — — X --^ X 

+ 1 ^ 


Ccmpare figure 3. 

With dT n -* X dT_ + -*-x dv" there results 
T It ■ y 


and "by siibatltutlon 


X At-. , + -Z£.x ^, 







. T ▼ "~: 

O + D ■ ■■ ■"— 




The Telocltj coiriponentB t 


ani T.^. for the new grid pclnt 
■ i/A 

ero known, according to those ccLLciJ-atlona, frcm ecLne.tlone (1 "b) and (3). 
Therefore- the air density also Is Imown accordlnf! to eqitatlon (2). 
Thus the Intezsiedlate region between the- subsonic flow with nr.wn 
perturlsatlon Telocltlos (calculation accordlnii to Premltl) and 
the 'boundaxy may Tae detexmlned end the position cf the sonic 
Telocity Ixjimdory and the Telocity direction alone that "boundary 
will Tie o^btalnod as final result. It will haTe to "bo further 
Investigated whether a calculation of the wliole s-jper sonic velocity 
field Tieyond the eonlc houndary up to the surface of the Txidy "by 
this method would "be practical. I!a the main, the time spent on 
the calculation H and the accuracy of the method which has "been 
developed here as contrasted with the metAiod of oliaracterlstlcs 
of Prandtl-BuBQBiaiin will haTe to "be considered. 

The difference method ^Aiich has "been cLeTelo?ed here does not, 
howerer, pexnlt a start of the numerical calculation at an arlaltrazy 
dlstanoe frcm the liody In the flow; for at a very great distance 
frcan the "body the calculation would not be euff lolently accurate 
since the moshes cf the grid v:<uld 1)0 too small. This connluslan 
can bo d.^avn from the foot tliat at a gireat distance the flow about 
any body without external forces can be replaced, except for "f"*^'''' 

ITACA mNo. ink 

deviations, "bgr the flov field around a single dlpole. It Is 
precisely tlie small derlatlons from the dlpole flov, lAiich exist 
even at the greatest distance, that pezmlt a continuation of the 
flov, for Instance according to the difference method, not a1x>ut 
a cylinder "but around the special profile. 

3. ^tgpersonlo Beglon 

I^ram the calculations descrllied &Taove, vhlch were carried 
out In the subsonic area according to Pran'ttl's rule, or hy the 
difference method, the hoiaidary line of the supersonic velocity 
field vas obtained. The Telocity along -Uils lino equals the 
sound Telocity and the dlr-ectlom ef the flov at every point of 
this line Is Imovn. The next problem consists In Joining the 
corresponding sitpersonlc velocity field to this boundary line; 
for Instance, by the method of oharewterl sties of Prandtl- 
Busemann. According to the method of characteristics it is 
Imovn that the changes of state occurring In a supersonic flov 
are manifested In e^cpanslon cr ccnpresslon vaves. Therefore, 
■vhen the flov passes throu^ thdtse lines of disturbance, the 
flov velocity and the flov diretttloii are modified by definite 
values 'Which can easily be detemlned from the graphical 
representation of the chai-acteristicsi® It can be Inferred fjxm 
the condition of constant velocity at the sonic boundaz? \ipjo 
that tvD vaves must originate at every point of the siExzU'c'^undary, 
namely an expansion and a compression vave. (See fig, h.) -Pie 
velocity direction Junips from one starting point of these waves 
at the sonic boundary to the next by a fixed amount for each case. 
Since these starting points of the vaves can be shifted in snj 
vay along the sonic velocity boundary, the beginning of a 
graphical representation of characteristics may be dravn for 
any given distribution of direction on the sonic boundary. 
Lllcevlse the starting points of the vaves may be shifted not 
only on the sonic boundary but also toirards points outside of 
the sonic boundary, so that the graphical representation of 
chai'acteristlcs nay be adjusted to any shape of the sonic 
boundary line. TherevLth it has been demonstrated that at least 
in certain cases there exists a possibility of continuing the 
subsonic flov Into a corresponding supersonic flov; finally there 
results the nev profile contour of the body in the flov/- 

^Compare L. Prandtl: "Fuhrer durch die StromungBlehre . " 
Verlag Vloveg and Sohn, Braunschvelg (19^2) S. 257 und folgende. 

*In order to render possible sufficiently accurate dravlngs 
of grids of characteristics for local supersonic velocity f leldjS, 
a graphical representation of charactoristics vith an interval 
of 1/5° has been completed at the STL; copies may be obtained. 

BACA OMiro. ink 

in. cALcncjfflioir op ah siamfee for a mixed 
SDEEaasoNic-saBigoiiic itxw 

A Blmple exsoDople will demanstrate the coAublnsd effect of the 
calculation method^ for various flow reeXariB, TOie velocity field 
for a two-dJmeiislanal Blonder "body vae determined In the proximity 
of the Ixjdy "by means of comformfll trwn af oimatlon and tlie associated 
distortion according to Prandtl for the Maoh number M » 0.86.- 
Thus -Uiere resulted a sonic velocity "boundary as Indicated In. 
figure 2(a). This sonic 'boundary, iftxloh vas ohtalned according 
to Prandtl*a method, could "be further litiproved "by continuing the 
flow step "by step to th^ Improved sonic velocity Twundnry; one 
would have to start from a "boundary lino with sufficiently antfin 
perturhatlon velocltios iistng the difference method descrihed 
ahove. Thio corrfictlon of the sonic velocity "boundary vaa at 
first diareiarded for the sako of simplicity, Elnco in c©neral, 
it can "bo noglectod for tho "basic calculatlona pltnned for this 
report. By means of the directions of velocity alone "^e sonic 
"boundary, the neb f,f characteristics for the Eiuporsonic flow can 
"be drawn so that the contour of the "body in the flow la a streami- 
line of this siiporscnic field. Only expansion waves start from 
the Bvirface of the "body, t&iile cantpresslon vaves, Tiilch are 
necessecry for a roconversion of the supersonic flew into a sub- 
sonic flow start witliout exception from the sonic "boundary. This 
"behaviour la an escontial criterion of the grid of characteristics. 
The manner of reconversion of the supersonic into a subsonlo flow 
(that is, for instance, \diether ateadlly or "by means of a com- 
pression shock) is doclsively Influenced "by the sonic vclocltj'- 

Of course, the contoiTT of the "body at the "boundary of the 
Bizpersonic field idiich has "been determined in the foregoing 
calculation of the example can not coincide with the contour of 
the original "body idilch was originally distorted according to 
Frandtl, since the supersonic field, "because of the expansion 
of the air, requires more space than the flow field ^ich was 
distorted according to Prandtl. Therefore the surface of the 
"body in the supersonic region will "be flatter than the surface 
of the original hody in order to satisfy this Incroeised need for 
space. The oixter su'bsonlc flow must "be adjusted to this increased 
need of space of the supersonic field In order to o"bteln -Qie 
original contour as the result of the calciilatian. This adjustment 
may "be made "by replacing the supersonic field with dlpoles or 
source-sin]): distrl"butlons, tdilch will widen tho stroamlines of 
the su"bsc(Qlc region originally ohtained "by Frandtl* s method. The 
strength of this source-sink distrl'butlon is Imown fj^Gsn the 
oc(ndltlcn that the space idilch is to "bo added has to cover tho 

8 IfACA. IMNo. 1114 

difference of the flov density In eupersonlo flov and flov 
according to Prandtl and that this space corresponds In 
magnitude to the flattening of the original "body *hlch resulted 
in figure 5(a). Bj- a stop-hy-step approximation the contour 
idilch results at the end of the culculatlon loay thus he 
adjusted to the desired ocntotcr. 

Hovever, the eTZonple treated here already permltc the 
ccncluplon that vbtxa. the local sound \elocity Is exceeded there 
exist solutions vlth reccnTorslcaa of the local s'aper sonic 
velocities into suhsonlc velocltloB vlthout conpi'eBslon shock, 
at least In frlctloolees flow. 

The pressure dlstrihiitlon >ailch wp.s fo'ind for the example 
treated.- Is congjared to the prensure 'iJ.sta.'l'bu.ttf-n of tnother 
slender "body (fig. 5(hJ tixldh was deteminod acc^i'dlnc oo the 
same lav of confozmal mapping and shculd a^^reo voll td-th the 
hody of the example. (Soe fig. 5(&).) ^^3 extriiune flitnesa 
of the pi-esQure dlstrlhutlon In the svpersorlc region in a 
special chai-acterlBtlc for the hody with local reclon of super- 
sonic velocity. This hehaviour coiTespmnds to tho sti-ong 
curvati-re of the contour of the "body imraodlately after the 
flov has ontered the supersonic region. D-jo to this large 
lorjal curvature, the hody 3 In this region vill produce even 
■for InccsqpresBible flov higher nege.tlve propsurea thpn the 
hody A. For this reason the two pressure dJ.strlbuticns Tdilch 
■♦•ere graphically rrprosentod cpnnot "no 'lireotlj ccoiparod. 


The representations of characteristics considered thus far 
vers notable for the fact that expansion or ccanpresslon vaves 
running in the same direction never intersect. (Ccsoiparo fig. 5(a).) 
An Intersecting of vaves running In the same diroctli..n is basically 
Impossible for expansion vaves since such 'VTUves 'jLLvoys diverge 
from their starting point. But canrpresalon wives oonvjrge; there- 
fort^ they could veil foim an envelope and cause a compression 

Such overlapplngs of vaves vere found for the sample body of 
figure 5(a) '^dien the free-strenm velocity vae inci^eased from 
M B 0.86 to H a 0.90. The ccnotructlon of the oxpaasian vaves 
vas started at the sonic boundary (In opposition to their actual 
direction) and already for the Mach number M a 0.86 (according 

TIACA TMNo. 232.k 

to figure ^(a)) led to a pressing together of the expansloti 
vaves near" the contoizfc'; for* the Mach ninaber M'b 0^90 - they 
ran Into each other even hefor^ reaching; the oontotir. ^Is 
condltlan Is physically senseless, that Is, a sjaanetrlcal 
solution for the sonic houndary iftilch iris found 'according to 
the Indicated scheme Is, at this Maoh numher M ■ 0^90, no 
longer posslhle. In this case the "expansion shock,'.' according 
to a consideration of Frandtl, rather will dissolve Into a 
group of divergent expansion waves orlg^jiatlng at the contour, 
vhlle the compression shuck idilch liialts the Erupersonlc area 
persists. Ill this manner one easily phtalns the flov picture 
found from many tosts for.'nblch the local sitpersonlc flov Is 
no longer reconveirted Into the su'bsonlc flov throu^ a contlnfjous 
phenomenon hut hy means of a cosapresslon shock. 

The preceding treatment vas hosed upon tho sonic velocity 
"boundary line idilch vas ohtolned according to Prandbl's rule 
without additional sou2'ce-3lnk todies wilch woiild have corre- 
sponded to the groater neei for space of the Buperaaalc reclon. 
The reeultlng solution tflilch was physically senselesH mi^t 
veil he caused hy the perhaps no longer appropriate conditions 
for the exlsteonce of the sonic velocity houndaay line. In order 
to ohteln a cleeirer vlev one consldei^s the llialtlng case ^ere 
thT expansion waves converge exactly at the contoirr. A flov 
around a coxner develops, as represented schemctlcally In 
flgirre 6; In tho s'.Tpersanlc reclon such a flow Is pooslble 
vlthout flow separations. Tbin aosunptlon and a given sonic 
velocity hotmdary offer the supersonic region tho opportizilty 
to fill the largest posslhle space without concave curvature of 
the hody In the Bupersonld domain. 

An admission of concave surfaces would presuppose a sonic 
velocity houndaiy ■which could no longer he produced from an 
Incompresslhle flov ahout a profile vlthout additional source 
and sink dlstrlhutlons In place of the supersonic region; or 
else one would have to drop the condition of f].ov sjonmetry. 
?lgio7e ^(a) demonstrate 8 clearly hov the surface always shows 
a convex curvatiUTs when the inccanlng wave Is a comprosslon wave, 
the outgoing wave an expansion wave. However, for the opposite 
condition of concave curvature In symmetrical flov tlie ccmlng 
In. of an expansion wave and going out of a compression w&ve 
would iresult In the amission of expansion waves from the sonic 
boundary In hoth cases'. But an expansion wave vhlch stairts at 
the sonic houndary cannot he obtained hy sl3iq;>ly exchanging the 
cantpresslon end expansion waves, for Instance according to the 
diagram In figure h; for there must always first appear an 
expansion wave and thun a compression wave In tho direction of 
the flov at the sonic velocity houndary, because the flov can 

10 KACA OMSo. lull- 

peisB flxaa the soolo tova&jeaj Into the sttperBonlc region onljr 
ty means of an expansion, ko. expansion vave stajrtlne kt the 
sonic botmdazy Is posslMe only for a certain shape of the 
comers In the graphical representation of characteristics: 
tiro expansion vaves are sent out from the comer BlTrmltwneov.aljr. 
(Ccaapeace fig. 7>) There Is, hovever, a condition; the sonic 
houndaiy In this region must talce a steeper course than the 
expansion vave vhlch Is Inclined tovard the direction of the 
flow 'by the Maoh angle. The strecmline through snich a comer 
point lies hetween the expansion -waves, 'Hhlle for a noimal comer 
It Is tangent to the apex of the triangle foimed liy the expansion 
and compresBloa -ware. 

It la an Important criterion for the comer vLth tvo expansion 
vsrea that the streamlines heoome steeper, not flatter, vLth 
Increasing distance from the tody; othei*wlse the flov -Hould not 
fit together after transition throu^t the two expansion wavas. 
This reemlt agrees vlth the fact that In a B7mr<jotrlcal supersonic 
region also the direction of the atreemllnes grows steeper with 
Increasing distance from the profile. The catise of this phencanenon 
Is that the stroaaillnes (hecause of the mflTHmnw flov density at 
Bonlo veloolty) are closest to each other at the sonic boundary 
vhlle there vlll he the greatest distance hetvoen them for the 
Btipersonlc region at the point of Tna-rlTmam velocity, that Is, 
generally, at the jKslnt of greatest thldmess of the hody. 

Such dletrlhutlons of slopes occur only In supersonic regions; 
they are not posslhle in suhsonlc areas for a flov ahout hodles 
vlth convex contotirs. Therefore it la not surprlslnn that the 
sonic velocity hoxmdary, \dilch vas obtained by distortion of the 
Incompressible flov about the body, vlll not load to a useful 
supersonic flov pattern as long as there are no singularities, 
as for instance source- slnlc distributions in the incompressible 
flov by means of lAiich the effect of the increased, expansion of 
the air in the supersonic region can be calculated. 

Presently a method is under investifjatlon in -vhich th? sonic 
velocity boundary may be adjusted to tho increasotl need for 
space of the supersonic flov by arrangement of singularities in 
the flov; the results of these investigations vlll soon be 
published separately. 


As stated in detail in the preceding chapter, concave body 
suz^'aces in the supersonic region are dependent t^pon the porticular 

BACA ON No. lllk 11 

type of comers In the net of ohaxaoterlstlcB -Nhlch le characterlEed 
"by tvo ezpansloD, vavee starting frcoa a point of ^ the soolc 'bottDdazT'. 
(Canipa3?e fig. 7.) The stlpulatian of flov STmMtry then causes 
compresslan waves to start from the corresponding points of the 
contour tdilch In turn cause these cnrnpresslon waves to proceed 
toward the sonic houndarj In pairs . Since these ccdZQresslon vaves 
'vbich Dtaiii at the contour alvays converge, such solutions ere 
posslhle only as long as the radiated coaiipresslaa vaves do not 
Intersect; othorvlBe a oonipresBlon shock vlll develop. Therefore 
the existence of STnanatrlcal flovs vlth local excpersonlc regions 
cannot "be counted upon for strongly concave curvatinres or for hlgjh 
Mach numbers, "because the ccsapresslon shoclc -would presuppose a 
symmetrical expansion shoclc In a corre°spondlng locatloa; li^ls 
ea^panslon shoclc, hovever, la not physically possllile. 

ISxcept for the special case descrl'bed aTx>ve, st^ersonlo flows 
on principle tend toward flow e;mnistry as will he shown In the 
exanrple represented In figure 8(a). Thore vlll alTrays develop a 
symmeti-lcal flow alang the wall with two chances In direction for 
IncQznpressl'ble flow without separations. It Is, however, lmo>qa 
that for pure supersonic flow the flow along the fTall Is unsynmetrlcaL 
(fig. 8(a)), If no additional guiding surfaces exist. Conditions 
of BTnmetry can "be achieved throu^^i calculations for the mixed 
BuTjoonlc- supersonic flow around this douhle comor: the outer 
subsonic flow at the sonic houndary will supply the missing 
synmetrlcal "boundaiy conditions (as for the exEmple In fig, 5(a)). 
But It does not appear Imposslhle that such a mixed superscnlc- 
BulsBonlc flow fdilch has "been made symmetrical "by the outer subsonic 
flov may he unstahle. Since the suhacnlc flov Is produced "by the 
form of the ohstacle In the' flow It t-jIII "bo symmetrical only for 
a sjonmetrlcal ohstacle. On the other hand the cuter flow will not 
"be Bynmietzlcal for an unsymmetrlcal obstacle, as for Instance a 
symmetrical "body with a local supersonic region and a can^tresslon 
ehock. Therefor^ It seems very doubtful whether the outer flow 
vbose fozra la decisively determined "by the foxm of the supersonic 
region, will In turn "be able to roEihape decisively the supersonic 
flov by forcing the symmetry conditions upon It.^ 

A oonf Iznatlon of the noneymnstry of the symnetrloal type of 
solution would explain the occurrence of unsjianetrlcal solutions 
with ccoDipresslon shoclcs idilch has been observed In tests. 

On this, occasion X should lUce to point otct that a ajmnetrlcal 
supersonic flov also could be suggested for the single wall represented 

^A solution of this stability problem shall be obtained by 
assuming a fgnan unsynnetrlcal defoznatlon of the sonic boundary and 
by then observing fAiether this defoznatlon Increases or dlmlnlElhes. 

12 m iTo. 1114 

in flsare 8(a) idilch Is ^bTsloally not real "but satisfies the 
conditions for potential flows vltb regard to contlntcLtj and 
Irrotatlonall ty as veil as Beznoulll's equation. This flotr Is 
represented jji figure SCb). Such a flow, aa Is vbU Imovn, Is 
phyBlcally real for the reason that a dlsturljance can not "be 
tranamltted upstream In supersonic flov. For solutions vhlch 
vero o1}talned gnalytlcftlly It must, therefore, alvays "be oFipeclally 
verified ^etber this additional condition has heen satisfied for 
the supersonic florf. 

71. SafMHT 

1. A method of calculating the approximate Telocity field 
for cortpresBlhl© fi-ows vlth local regions of supersTnlc velocity 
has heon presented. Starting from the flov at a lar^ distance 
from the profile deteimlned' according to Franltl's rule, this 
outer flov was continued t-o the sonic Telocity "boundary ty means 
of a numerical methcd; the metiiod ct characteristics of Frandtl- 
Busemann was applied for continuation "beyond that "boundary. These 
calculations result finally In the contour of the profile In the 
region of supersonic Telocity. 

2. It has "been deinonstrated In an exazople that mixed subsonic- 
supersonic flows a'bcut two-dimensional "bodlos can "be calculated 
tdiere net only the transition fi-om su'bsonlc to supersonic "but also 
the transition from. s''rpersonlc to su'bsonlc taJces place continuously, 
that Is, "vTlthout pressure Jump. 

3. ITo physically real local area of supersonic Telocl'ty could 
"be determined for the example considered here -vihen the sound 
Telocity was far exceeded "because then esqpanslon or scmpresslon 
shocks occurred. Howrarer, there Is a prospect of calculating mixed 
flovB for such cases also: "before starting the calculation one 
would hare further to extend the oiiter su'bsonlc flow "by means of 
source and sink distributions; In this way tJie greater need, for 
sp£u;e of the supersonic Telocity field would "be satisfied. 

l^. The symmetry of the local area of supersonic Telocity for 
mixed supersonic- su'bsonlc flows may "be enforced "by the outer 
su'bsonlc flow. Since, hoverer, the outside subsonic flow In 
turn must "be produced "by the profile In the flow or "by "the local 
area of supersonic Telocity, the assuioptlon seems Justified that 

BAGfL OMNo. 111b 13 

often only an Tgista'ble BQnraetrloal flov can "be pcroduced "by the 
outolde subsonic flov; and this loistable Bjnmetrlcal flov vlll 
turn even at email dlsturljances Into the unEywmotrloal case vlth 
cceoopresBlon efhooks. 

Translated "by M&ry L. Mahler 
ITatlotoal Mrlsoxy Ccemnlttee 
for Aoronautlos 


Sonic velocity-Boundary 


Figure 1. Schematic representation of the flow regions for mixed subsonic 
-supersonic flow. 

Fig. 2 

NACA TM No. 1114 

Figure 2. Partition of the flow field for the difference method. 

NACA TM No. 1114 

Figs. 3,4 



via.'' Z,6 

Figure 3. Relation between air density and velocity under the assumption 
of an adiabatic change of state. 

re gi on 

Sonic velocity-Boundary 

Direction of velocity 


Figure 4. Sonic boundary with the field of expansion and compression waves, 

Sonic velocity-Boundary 

Expansion waves 

Compression waves 

Direction of the 
oncoming flow 



Direction jump in the net of characteristics 



Figure 5a. Grid chacteristics in the local supersonic region for a two-dimensional body 
B (relation of thickness d/1 = 0.0715 at the Mach number M = 0.86. 





A: Pressu 
at rfec 

re distribution 
i-number M= 


.-*j — 











IT ^ 

\ — \ — 1 — \ — r — r 

0,2 H/ 1,1 0,1 1,0 






Promidli'-om ) 

Figure 5b. Calculated pressure distribution for two two-dimensional bodies in incompressible 
flow and in compressible flow with local area of supersonic velocity. 

Figs. 6,7 

NACA TM No. 1114 

Expansion waves 

Sonic velocity-Boundary -^,/x 

Compression waves 

Direction of velocity 

Figure 6. Mixed subsonic-supersonic flow about a body with plane surface in 
the supersonic region. 

Normal corners 

Direction of the 
oncoming flow 

Sonic velocity-Boundary 

Expansion wave 
Compression wave 

Figure 7. Schematic representation of the two different kinds of wave 
radiation at the sonic velocity boundary. 

NACA TM No. 1114 

Fig. 8 

Velocity- Vector 


Figure 8a. Supersonic flow about a double comer. 


Expans i on 



Figure 8b. Symmetrical potential flow with supersonic free-stream 

velocity about a physically not realizable double corner. 

NASA Technical Ubrai 

3 1 

76 01441